This work started w/ my struggle to understand fault-tolerant cluster states, and in particular why they always magically implement the same QEC problem as non-MBQC circuits. Understanding this correspondence as chain complex equivalence finally solved it for me, and I hope for other people too :)

Amazing !! Thank you!
Congratulation on the manuscript!

Now, if you ask the question the other way of how many circuit constructions would do that, there might be only this one?